I was excited to find a new Illustrative Mathematics task using coordinate geometry.
CCSS-M G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
In the picture below a square is outlined whose vertices lie on the coordinate grid points:
The area of this particular square is 16 square units. For each whole number n between 1 and 10, find a square with vertices on the coordinate grid whose area is n square units or show that there is no such square.
As a precursor to the task, I included the following statements on the bell work for students to discuss in their groups before we had a brief class discussion.
The three sides of a right triangle can all be even.
The three sides of a right triangle can all be odd.
Last year, I heard Linda Griffith talk about giving a part of this task to some students in Arkansas. I began with my students the way she began with hers. Each student had a sheet of graph centimeter graph paper and a straightedge. Near the top left corner, draw a square with an area of 1 square centimeter.
Challenge accepted, although some students drew their square in the top right corner.
Next, I want you to choose a point, which can be above your square on even on your square, and I want you to dilate your square by a scale factor of two.
This took longer, but it was a good reminder of what we need for a dilation.
What happened? What can you tell me about your image?
It has an area of 4 square centimeters.
How do you know?
I counted the squares.
Someone else noted that the similarity ratio is 1:2, so the ratio of the areas is 1:4.
What will happen if you dilate your original square by a scale factor of 3.
We will get a square with an area of 9.
And so they did.
Now. Here is our goal for this lesson: For each whole number n between 1 and 10, find a square with vertices on the coordinate grid whose area is n square units or show that there is no such square.
So far, we have 1, 4, and 9. What do you know about those numbers?
They’re perfect squares.
Yes. So now I want you to draw a square with an area of 2 square centimeters. I’d like for you to work by yourself for 2 minutes, and then you can share what you’ve found with your group.
I watched while they worked. I saw many students approximating √2 on their calculator. I saw several students who had made a rectangle with an area of 2 square centimeters. I saw one student who had immediately thought of 45-45-90 triangles and had drawn a square with an exact area of 2. Everyone was doing something, even if they were using approximations.
After two minutes, I told students they could work together now, and that I had two reminders: I have asked you to draw a square. And I want it to have an exact area of 2 square centimeters.
I heard great conversation. I asked a few of those who had approximated the side length of their square how they knew the side was √2. Linda Griffith told a great story last year about some of her students: they decided to put “not drawn to scale” next to their diagram, as they had seen on one too many of the diagrams from their geometry class. Several others made the 45-45-90-connection for an isosceles right triangle with a leg of length 1 cm to get the desired square. I listened to one group who realized they had confused whether a square is always a rectangle or a rectangle is always a square take their rectangle and compose its parts a different way to get a square.
I decided to have them share first. It occurred to me after they started talking that I should video their explanation. I caught part of it.
I love that these two took their rectangle of area 2 and rearranged it to make a square of area 2.
Next I asked the student who had immediately thought of 45-45-90 to explain her thinking.
She related her work to the Pythagorean Theorem.
And finally one other student shared who had composed his square differently from the girls with the rectangle.
Now that we have a square with an area of 2, what other square areas can we easily get?
Of course a dilation by a scale factor of 2 will give us a square with an area of 8.
What side length does that square have? 2√2.
So what is next? We still need squares with areas of 3, 5, 6, 7, and 10.
What could we do to get 5?
Several students simultaneously thought about 3-4-5 right triangles. So what does that give us? An area of 25, which we can get with oblique side lengths from the 3-4-5 triangle or horizontal/vertical side lengths of 5 cm.
It isn’t really 5 we need. What can we do to get √5 for a side length?
Students continued working, many coming up with a 1-2-√5 triangle from which to draw a square with an area of 5.
If we can get 5, can we get 10?
I was expecting to hear 12+32=10, and I did hear that. But I also heard (√5)2+(√5)2=10, which I didn’t hear as loudly because I wasn’t expecting to hear it. You would think I’d have learned by now to pay closer attention to what my students actually say. What I am learning, though, is that it takes time to process student thinking for a task that isn’t “cookie cutter”, and I don’t always do that quickly in class, especially when the bell is about to ring. We ended with a discussion of more than one way to get a square with an area of 10 – and I left 3, 6, and 7 for the students to finish exploring outside of class.
So I would have liked to talk about why 3, 6, and 7 don’t work. We didn’t get there this year.
But we did make it farther than last year as the journey continues …